Triangle singularity in $B^{0}\to\pi^{-}K^{+}X(3872)$ via the $D_{s1}\bar{D}D^{*}$ loop and possible precise measurement of the $X(3872)$ mass

The $B^{0}\to\pi^{-}K^{+}X(3872)$ is one of the reactions where the $X(3872)$ is discovered, of which the branching fraction is around $(2.1\pm 0.8)\times 10^{-4}$ [24, 25]. We notice that this process may receive contributions from the triangle diagram displayed in Fig. 1. In this rescattering process, the $B^{0}\to\pi^{-}D_{s1}(2536)\bar{D}^{0}$ is a Cabibbo-favored decay, and the $D_{s1}(2536)$ mainly decays into $D^{*}K$. Therefore we can expect this rescattering may play a role in $B^{0}\to\pi^{-}K^{+}X(3872)$. Furthermore, the intriguing feature of this rescattering process is that the three intermediate particles can be (nearly) on-shell simultaneously in some kinematic regions, and a TS located close to the physical boundary in the complex energy plane of the amplitude can develop from this $D_{s1}\bar{D}D^{*}$ loop. As a result the transition amplitude of $B^{0}\to\pi^{-}K^{+}X(3872)$ will be enhanced in some areas and a TS peak can be expected to arise in the $K^{+}X(3872)$ invariant mass spectrum.

Assuming the $X(3872)$ mass $m_{X}$ is not fixed, the location of the TS in the $m_{X}$ or $M_{KX}$ complex plane can be determined by solving the Landau equation [26, 27]. In terms of Eqs. (3) and (4) of Ref. [28] derived from the Landau equation and a dispersion analysis, we can obtain the TS window corresponding to Fig. 1:

where the central mass values of relevant mesons from Ref. [2] are adopted. The meaning of the above window is : When $m_{X}$ increases from $3871.69$ MeV, i.e., the ${D}^{*0}\bar{D}^{0}$ threshold, to $3875.24$ MeV, the TS in $M_{KX}$ moves from $4403.66$ to $4399.93$ MeV; Vice versa, when $M_{KX}$ increase from $4399.93$ MeV, i.e., the $D_{s1}(2536)\bar{D}^{0}$ threshold, to $4403.66$ MeV, the TS in $m_{X}$ moves from $3875.24$ to $3871.69$ MeV. We also refer to Refs. [27, 26, 29, 30, 28] for more detailed discussion on the locations of the TS in various kinematic configurations.

For the $D^{*}\bar{D}^{*}D$ loop mentioned in the introduction, ignoring the $D^{*}$ width, when $\delta_{X}=0$, the TS position of $M_{\gamma X}$ is about 2.7 MeV larger than the $D^{*}\bar{D}^{*}$ threshold, while that of $M_{\pi X}$ is 0.3 MeV. For the $D_{s1}\bar{D}D^{*}$ loop, ignoring the $D_{s1}(2536)$ width, when $\delta_{X}=0$, the TS position of $M_{KX}$ is 3.7 MeV larger than the $D_{s1}\bar{D}^{0}$ threshold. Larger gap between the TS position and pertinent threshold indicates that the line shape could be more sensitive to the $X(3872)$ mass compared with the $D^{*}\bar{D}^{*}D$ loop. Besides, the charm-strange meson $D_{s1}(2536)$ is also very narrow. The PDG average value is $\Gamma(D_{s1}(2536)^{\pm})=0.92\pm 0.05$ MeV [2]. The narrowness of intermediate particles in the triangle diagram is one of the key reasons why the TS mechanism of measuring the $X(3872)$ mass may work.

When employing the TS mechanism to determine the $X(3872)$ mass, we are interested in the line shape of $M_{KX}$ distributions in the vicinity of $D_{s1}(2536)\bar{D}^{0}$ threshold, therefore we only take into account the amplitude involving the lowest angular momentum between $D_{s1}(2536)$ and $\bar{D}^{0}$. The non-relativistic amplitude of $B^{0}\to\pi^{-}D_{s1}\bar{D}^{0}$ reads

with $\vec{P}=-\vec{p}_{\pi}+\vec{p}_{B}=0$ in the $D_{s1}(2536)\bar{D}^{0}$ c.m. frame. The coupling constant $\mathcal{C}_{1}$ can be determined from the experimental data. However, the branching fraction of $B^{0}\to\pi^{-}D_{s1}(2536)\bar{D}^{0}$ is not known yet. The experiment gives $\mathcal{B}(B^{0}\to D_{s1}(2536)^{+}D^{*-})\times\mathcal{B}(D_{s1}(2536)^{+}\to(D^{*0}K^{+}+D^{*+}K^{0}))=(5.0\pm 1.4)\times 10^{-4}$ [2]. Assuming the $\pi^{-}\bar{D}^{0}$ states in $B^{0}\to\pi^{-}D_{s1}(2536)\bar{D}^{0}$ are fully from the $D^{*-}\to\pi^{-}\bar{D}^{0}$ decays, we estimate the coupling constant $\mathcal{C}_{1}\approx 7.0\times 10^{-6}$ using the experimental central values.

The $D_{s1}(2536)$ mainly decays into $D^{\ast}K$ in relative S-wave, and the amplitude reads

where the coupling $g_{D_{s1}D^{*}K}$ can be determined from the experimental data. Using the central values of the particle masses and branching fraction from Ref. [2], we obtain $g_{D_{s1}D^{*}K}\approx 0.78\ \mbox{GeV}$. Actually, this $S$-wave decay mode is supposed to be suppressed by the heavy quark spin symmetry (HQSS). On the other hand, the HQSS is preserved for the $D^{*}K$ $D$-wave decay mode, but this mode is highly suppressed by the limited phase space, since the $D^{*}K$ threshold is rather close to the $D_{s1}(2536)$ mass. These reasons lead to that the $D_{s1}(2536)$ is so narrow, and we can take advantage of this characteristic to make the TS mechanism work. For another charmed-strange meson $D_{s2}(2573)$, although it can also decay into $D^{*}K$ and its mass is just a little larger than that of $D_{s1}(2536)$, its width is about 16.9 MeV, which is much larger than $\Gamma_{D_{s1}(2536)}$. We therefore do not take into account the contribution from the $D_{s2}\bar{D}D^{*}$ loop in this work.

For the fusion of $\bar{D}^{0}$ and $D^{*0}$ into the $X(3872)$, the amplitude can be written as

Supposing the $X(3872)$ is a pure hadronic molecule, the coupling $g_{X}$ can be estimated by using the Weinberg compositeness condition [31, 32, 33], which gives

where $\delta_{X}$ is the binding energy, and $\mu$ is the reduced mass of the $\bar{D}^{0}$ and $D^{\ast 0}$, i.e., $\mu=m_{\bar{D}^{0}}m_{D^{*0}}/(m_{\bar{D}^{0}}+m_{D^{*0}})$. The above equation is valid for the bound state ($\delta_{X}>0$). For the resonant case, the coupling can be evaluated as the residue of the $\bar{D}D^{*}$ scattering $T$ matrix [34]. The coupling $g_{X}$ only affects the strength of the rescattering amplitude but will not change the line shape of the distribution curve. We therefore take a moderate value $g_{X}=3~{}\rm{GeV}$ which corresponds to the $\delta_{X}$ is at the order of magnitude of 100 keV, as did in Ref. [22]. We should also mention that, although the $g_{X}$ does not affect the line shape behavior, it intrinsically depends on the nature of $X(3872)$.

The decay amplitude of $B^{0}\to\pi^{-}K^{+}X(3872)$ via the $D_{s1}\bar{D}D^{*}$ triangle loop figured in Fig. 1 reads

For the spin-1 state, the sum over polarization takes the form $\sum\epsilon_{i}\epsilon_{j}^{*}=\delta_{ij}$. It should be mentioned that we adopt the non-relativistic amplitudes in Eqs. (4), (5) and (6), but we do not take the non-relativistic approximations for the denominators of the three propagators in the loop integral as shown in Eq. (II.2), since the formalism of these vertexes does not affect the line shape behaviour around the threshold we are interested in. The line shape of the distribution curve mainly depends the loop integral, which is numerically evaluated by employing the program package LoopTools [35].

The partial decay width of $B^{0}\to\pi^{-}K^{+}X(3872)$ reads

where

with $\lambda\left(x,y,z\right)=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$.

The TS is a logarithmic singularity. To avoid the infinity of the loop integral in the physical region, one can replace the Feynman’s $i\epsilon$ for the propagator by $im\Gamma$ with $\Gamma$ the total decay width, or equivalently replace the real mass $m$ by the complex mass $m-i\Gamma/2$, which will remove the TS from the physical boundary by a small distance [36, 37, 38]. The physical meaning of this complex mass prescription for avoiding the infinity is obvious. As long as the kinematic conditions for the TS being present on the physical boundary are fulfilled, it implies that the intermediate state (here is $D_{s1}(2536)$) is unstable, and it is necessary to take the finite width effect into account. Correspondingly, we replace the mass $m_{D_{s1}}$ in Eq. (II.2) by $m_{D_{s1}}-i\Gamma_{D_{s1}}/2$. The central values $m_{D_{s1}}=2535.11$ MeV and $\Gamma_{D_{s1}}=0.92$ MeV from Ref. [2] are adopted in the numerical calculations.

The invariant mass distributions of $K^{+}X(3872)$ via the triangle diagram in Fig. 1 are displayed in Fig. 2. The $X(3872)$ mass is varied in a window $\delta_{X}\in[-150,\ 150]$ keV. One can see that for different $m_{X}$ or $\delta_{X}$, the line shapes are also quite different. For every distribution curve in Fig. 2, there is a cusp just at the $D_{s1}(2536)\bar{D}^{0}$ threshold. But these threshold cusps are smeared to some extent by the width effect of $D_{s1}(2536)$. The peak looks more clear and narrower for the negative $\delta_{X}$ compared with that for the positive $\delta_{X}$. Supposing the masses of intermediate states are real, if $m_{X}$ is larger than or equal to the $D^{*0}\bar{D}^{0}$ threshold ($\delta_{X}\leq 0$), the TS in $M_{KX}$ can be present on the physical boundary, and the corresponding TS peak in the distribution curve can be very sharp and the peak position is a little bit higher than the $D_{s1}(2536)\bar{D}^{0}$ threshold, but if $m_{X}$ is smaller than the $D^{*0}\bar{D}^{0}$ threshold ($\delta_{X}>0$), the conditions of TS in $M_{KX}$ being present on the physical boundary can never be fulfilled, and one does not expect a sharp peak to appear in the distribution curve. Just because of this special character, the line shapes of $M_{KX}$ spectrum are very sensitive to the $m_{X}$. Especially, one can easily distinguish whether the $\delta_{X}$ is positive or negative by measuring the $K^{+}X(3872)$ spectrum.

For the peaks shown in Fig. 2, if we define a “width” at half maximum of the line shape, we can see the width can be as large as 3 to 5 MeV. Although this width is not a well defined quantity because of the asymmetric line shape, in terms of which we can still see an advantage for experiment: the larger width of the TS peak may reduce the requirement for energy resolution in measuring the line shape. The larger width is related to the larger TS window as shown in Eq. (3).

The $X(3872)$ is not a stable particle, and we need to take into account its width impact on the $KX(3872)$ line shape we are interested in. The Belle collaboration reports the branching fraction $\mathcal{B}(X(3872)\to D^{0}\bar{D}^{*0})$ is about $37\%$ [39], and the $\mathcal{B}(X(3872)\to D^{0}\bar{D}^{0}\pi^{0})$ is about $40\%$ with large uncertainties [40]. The partial decay width $\Gamma(X(3872)\to D^{0}\bar{D}^{0}\pi^{0})$ is expected to be about 40 keV in literatures [41, 42, 43]. Then the total width of $X(3872)$ can be estimated at around 100 keV. On the other hand, in Ref. [44], the LHCb collaboration reports the Breit-Wigner (BW) width of $X(3872)$ is $\Gamma_{BW}=1.39\pm 0.24\pm 0.10$ MeV. But considering that the proximity of $X(3872)$ mass to the $D^{0}\bar{D}^{*0}$ threshold may distort the line shape from the simple BW form, the LHCb also reports the full width at half maximum (FWHM) of the line shape is $\Gamma_{FWHM}=0.22^{+0.07+0.11}_{-0.06-0.13}$ MeV by using a Flatté-inspired model [44]. One can see that the width value is subtle and highly depends on the fitting methods for the near threshold state $X(3872)$. The PDG 2022 gives the averaged BW width $\Gamma_{X}=1.19\pm 0.21\ \mbox{MeV}$ [2]. We take into account the $X(3872)$ width impact on the $KX$ line shape by introducing a new invariant mass distribution function

where the spectral function $\rho_{X}$ is defined to be

with

The same functions are adopted in Ref. [22]. Another spectral function with Flatté parametrization is adopted in Ref. [18]. The corresponding new distribution curves are displayed in Fig. 3. Compared with Fig. 2, we can see that the strengths are weakened and the curves are smoothed to some extent. When $\Gamma_{X}$ is set to be 100 keV and 220 keV, one can still see relatively larger discrepancy between the curves corresponding to negative and positive $\delta_{X}$. But when $\Gamma_{X}$ is set to be 1.19 MeV, the discrepancy between different curves is tiny and those curves nearly overlap with each other, which implies the sensitiveness of the line shape on $\delta_{X}$ is reduced when $\Gamma_{X}$ is larger. From this point of view, we comment that if the width of $X(3872)$ is as large as 1 MeV, the TS mechanism of measuring its mass may be ruined.

If we use the TS mechanism to measure the discrepancy between $m_{X}$ and $D^{*0}\bar{D}^{0}$ threshold, the $X(3872)$ needs to be reconstructed in decay modes other than the $D^{0}\bar{D}^{0}\pi^{0}$, which can be $J/\psi\pi^{+}\pi^{-}$, $J/\psi\pi^{+}\pi^{-}\pi^{0}$ and so on. Otherwise one has to take into account the interference term between the rescattering triangle diagram and the cascade decay process $B^{0}\to\pi^{-}D_{s1}(2536)^{+}\bar{D}^{0}\to\pi^{-}K^{+}D^{0}\bar{D}^{0}\pi^{0}$. This interference is subtle and the tree level cascade decay process cannot be treated as a smooth background near the TS regions because of the so-called Schmid theorem [45, 46, 47, 48, 49]. This point has ever been pointed out in Ref. [17].